I am looking for software to create an animated video like this:or an even better: http://vimeo.com/63457188. It would be great if you could share more than one tips. Are there any such free apps out there?
The animation that you refer to was probably made with Adobe After Effects. You can animate this kind of thing for 'free', if you don't value your time, using PhotoShop CS2 (free from Adobe) and the Quicktime player which will import image sequences and save them as movies.
But your hope for free software misses the point that the costly parts of animation are time and skill. More expensive tools frequently have built-in time savers. After Effects is certainly one of those. It won't save you, however, from having to get skill.
Synfig Studio is a free option. It's a free 2d vector animation package, similar to Flash. I haven't used it, so I can't vouch for how easy it is to learn.
Since Blender can pretty much do anything - provided you can work out how to use it - you could probably do this in Blender. Though it's not really purpose-built for 2d animation or motion graphics.
For some reason there aren't many other free / FOSS motion graphics tools out there. The Jah Shaka project was an attempt to get a FOSS tool similar to for After Effects off the ground, but the project went belly up. It may rise from the ashes some time in the future, they seem to have a comeback every few years and then disappear again, but you never know.
A low-price option if you have a mac is Motion. Only low-price if you already have the mac though.
Any of these options will require weeks at least of learning time (I've been doing it twenty years and still learn something every day), so unless you want to pursue it as a career your cheapest option would be to hire a freelancer.
This type of animation is known as motion graphics. It is generally done with keyframe animation and vector graphics. It unfortunately is not something that has many free tools available however and it does require developing a fair bit of experience and quite a lot of time to do well.
As Craig mentioned, After Effects is really the ideal package for this kind of thing. Flash is also useful. You may also find yourself using Illustrator for actually building some of the vector graphics. You can use raster graphics instead (Photoshop) but the quality of zooms and rotations and such is not as high with raster graphics as it is with vector.